Abstract

The main object of this paper is to establish the classification and some criteria of the limit cases for singular second-order linear equations with complex coefficients on time scales. According to the number of linearly independent solutions in suitable weighted square integrable spaces, this class of equations is classified into cases I, II, and III. Moreover, the exact dependence of cases II and III on the corresponding half-planes is given and some criteria of the limit cases are established.

where p and q are complex-valued rd-continuous functions, w is a real rd-continuous function; \(p(t)\neq0\) and \(w(t)>0\) for all \(t\in [\rho(0),+\infty)\cap\mathbb{T}\); \(p^{-1}\) is Δ-integrable on \([\rho(0),+\infty)\cap\mathbb{T}\); \(\lambda\in\mathbb{C}\) is the spectral parameter; \(\mathbb{T}\) is a time scale with \(\rho(0)\in\mathbb{T}\) and \(\sup\mathbb{T}=+\infty \); \(\sigma(t)\) and \(\rho(t)\) are the forward and backward jump operators in \(\mathbb {T}\); \(y^{\Delta}\) is the Δ-derivative; and \(y^{\sigma}(t):=y(\sigma (t))\). In general, equation (1.1) is formally self-adjoint if and only if \(p(t)\) and \(q(t)\) are real, equation (1.1) is called formally non-self-adjoint when \(\Im p(t)\neq0\) or \(\Im q(t)\neq0\).

In 1910, Weyl gave a dichotomy of the limit-point and limit-circle cases for singular formally self-adjoint second-order linear differential equation [1]. Later, Titchmarsh, Coddington, Levinson et al. developed his results and established the theory of Weyl-Titchmarsh [2, 3]. Their work was further developed to higher-order differential equations and continuous Hamiltonian systems [4–9]. Singular spectral problems of self-adjoint scalar second-order difference equations over infinite intervals were first studied by Atkinson [10]. His work was followed by Hinton, Jirari et al. [11, 12]. Further, their work has been developed to formally self-adjoint Hamiltonian difference systems [13, 14]. In the past few years, the theory of Weyl-Titchmarsh has been greatly developed and generalized to the discrete symplectic systems. Many important results have been established [15–17].

In 1957, Sims obtained an extension of the Weyl classification for formally non-self-adjoint second-order linear differential equations, which the leading coefficients are identical to 1 and the potential functions are complex [18]. Later, Brown et al. in 1999 [19] extended this work to a more general case:

where \(-\infty< a< b\leq+\infty\), p and q are complex-valued functions, w is a weight function, \(p(t)\neq0\) and \(w(t)>0\) a.e. \(t\in[a,b)\), \(p^{-1}(t)\), \(q(t)\), and \(w(t)\) are locally integrable on \([a,b)\), λ is a spectral parameter. They divided the equations into three cases by using the m-function, which was defined on a collection of rotated half-planes. Recently, non-self-adjoint Sturm-Liouville difference equations and Hamiltonian difference systems have been discussed [20, 21]. Especially, Wilson in [21] gave a discrete analog of the work of Brown et al. [19] on the following non-self-adjoint second-order difference equations:

where \(\mathbb{N}_{0}=\{0,1,2,\ldots\}\), Δ is the forward operator, i.e., \(\Delta x(n)=x(n+1)-x(n)\); \(p(n)\) and \(q(n)\) are complex numbers, and \(w(n)\) is a real number; \(p(n)\neq0\) for \(n\in\{ -1\}\cup\mathbb{N}_{0}\) and \(w(n)>0\) for \(n\in\mathbb{N}_{0}\); λ is a spectral parameter. He also divided the equations into three cases, by using the m-function. The classification for formally non-self-adjoint second-order differential or difference equations is related to the corresponding half-planes. But in [19, 21], the authors did not discuss whether there existed the case where the equation was in the case II with respect to a rotated half-plane and in case III with respect to another one. More recently, Qi, Zheng, and Sun [22–24] proved that case II and III depend on the corresponding half-planes by illustrating two examples and gave two propositions to show how case II and III depend on the corresponding half-planes.

In the past 20 years, a lot of efforts have been made in the study of regular spectral problems on time scales [25–32]. But singular spectral problems have started to be considered only quite recently [33–40]. In 2012, we employed Weyl’s method to divide the following formally self-adjoint second-order linear equations on time scales into limit-point and limit-circle cases [38]:

where \(p^{\Delta}\), q, and w are real and piecewise continuous functions on \([\rho(0),+\infty)\cap\mathbb{T}\), \(p(t)\neq0\) and \(w(t)>0\) for all \(t\in[\rho(0),+\infty)\cap\mathbb{T}\), \(\lambda \in\mathbb{C}\) is the spectral parameter. It has been found that the formally non-self-adjoint second-order linear differential equations (1.2) and difference equations (1.3) can be both divided into three cases, by using the Weyl method. We wonder whether it holds on time scales. The main purpose of this paper is to extend the pioneering work of classification of (1.2) and (1.3) to equation (1.1), present the exact dependence of cases II and III on the corresponding half-planes, and establish several criteria of the limit cases for equation (1.1).

The rest of this paper is organized as follows. In Section 2, some basic concepts, fundamental theories, and propositions are introduced. In Section 3, a family of nested circles which converge to a limiting set is constructed. The classification of the limit cases and the exact dependence of limit cases on the corresponding half-planes are given. Finally, several criteria of the limit cases are established in Section 4.

2 Preliminaries

In this section, first, we introduce some basic concepts and fundamental results on time scales.

Let \(\mathbb{T}\subset\mathbb{R}\) be a nonempty closed set. The forward and backward jump operators \(\sigma,\rho:{\mathbb{T}}\to \mathbb{T}\) are defined by

respectively, where \(\inf\emptyset=\sup\mathbb{T}\), \(\sup\emptyset =\inf\mathbb{T}\). A point \(t\in\mathbb{T}\) is called right-scattered, right-dense, left-scattered, and left-dense if \(\sigma (t)>t\), \(\sigma(t)=t\), \(\rho(t)< t\), and \(\rho(t)=t\), separately. Denote \(\mathbb{T}^{k}:=\mathbb{T}\) if \(\mathbb{T}\) is unbounded above and \(\mathbb{T}^{k}:=\mathbb{T}\setminus(\rho(\max\mathbb{T}),\max \mathbb{T}]\) otherwise. The graininess \(\mu:{\mathbb{T}}\to [0,+\infty)\) is defined by

$$\mu(t):=\sigma(t)-t. $$

Let f be a function defined on \(\mathbb{T}\). f is said to be Δ-differentiable at \(t\in\mathbb{T}^{k}\) provided there exists a constant a such that for any \(\varepsilon>0\), there is a neighborhood U of t (i.e., \(U=(t-\delta, t+\delta)\cap \mathbb{T}\) for some \(\delta>0\)) with

In this case, denote \(f^{\Delta}(t):=a\). If f is Δ-differentiable for every \(t\in\mathbb{T}^{k}\), then f is said to be Δ-differentiable on \(\mathbb{T}\). If f is Δ-differentiable at \(t\in\mathbb{T}^{k}\), then

A function f defined on \(\mathbb{T}\) is said to be rd-continuous if it is continuous at every right-dense point in \(\mathbb{T}\) and its left-sided limit exists at every left-dense point in \(\mathbb{T}\). The set of rd-continuous functions \(f:{\mathbb{T}}\rightarrow{\mathbb {R}}\) is denoted by \(C_{\mathrm{rd}}({\mathbb{T}})=C_{\mathrm{rd}}({\mathbb {T}},{\mathbb{R}})\). The set of kth Δ-differentiable functions with rd-continuous kth derivative is denote by \(C^{k}_{\mathrm{rd}}({\mathbb {T}})=C^{k}_{\mathrm{rd}}({\mathbb{T}},{\mathbb{R}})\).

where \(\overline{\operatorname{co}}\) denotes the closed convex hull. For \(\lambda_{0}\in\mathbb{C}\setminus Q\), denote by \(K=K(\lambda_{0})\) its nearest point in Q and denote by \(L=L(\lambda_{0})\) the tangent to Q at K if it exists, and otherwise any line touching Q at K. We then perform a transformation of the complex plane \(z\mapsto z-K\) and a rotation through an angle \(\eta=\eta(\lambda_{0})\in(-\pi,\pi]\), so that the image of L coincides with the imaginary axis. Furthermore, the images of \(\lambda_{0}\) and the set Q lie in the negative and non-negative half-planes, respectively. In other words, for all \(t\in [\rho(0),+\infty)\cap\mathbb{T}\) and \(r\in(0,+\infty)\),

where \(\alpha\in\mathbb{C}\). Since their Wronskian is identically equal to −1, these two solutions form a fundamental solution system of (1.1). We form a linear combination of \(y_{1}(t,\lambda)\) and \(y_{2}(t,\lambda)\)

The critical point of (3.6) is \(\tilde{z}=-\frac{e^{i\eta }p(b)y_{2}^{\Delta}(b,\lambda)}{y_{2}(b,\lambda)}\), and we require this point to be such that \(\Re[\tilde{z}]\) is negative. Upon calculation, we find that

Hence, by (2.2) and (2.4), \(\Re[\tilde{z}]<0\) as required. Therefore, when (2.4) is satisfied, \(z\mapsto m_{b}(\lambda,z)\) maps \(\Re [ze^{i\eta}]\geq0\) onto a closed disc \(D_{b}(\lambda)\). By the properties of this transformation, the center \(a_{b}(\lambda)\) of the disc \(D_{b}(\lambda)\) corresponding to the reflection of the critical point in the imaginary axis. Therefore,

Proof

It follows from Theorem 3.1 that \(m=m_{b}(\lambda,z)\in D_{b}(\lambda)\) if and only if \(\Re[e^{i\eta}z_{b}(\lambda,m)]\geq0\), that is, \(\Re [e^{i\eta}p(b)y^{\Delta}(b,\lambda,m)\overline{y(b,\lambda,m)}]\leq0\). As in (3.7), \(\Re[e^{i\eta}p(b)y^{\Delta}(b,\lambda,m)\overline {y(b,\lambda,m)}]\leq0\) can be written as

Hence, \(D_{b_{2}}(\lambda)\subset D_{b_{1}}(\lambda)\). That is, the discs \(D_{b}(\lambda)\) are nested as \(b\rightarrow+\infty\). This completes the proof. □

Corollary 3.1

For\(\lambda\in\Lambda_{\eta ,K}\), as\(b\rightarrow+\infty\), the discs\(D_{b}(\lambda)\)contract either to a disc\(D_{\infty}(\lambda)\)or to a point\(m(\lambda)\). These represent limit-circle and limit-point cases, respectively.

This result enables us to give the following full characterization of equation (1.1).

Definition 3.1

Let \((\eta,K)\in S(\alpha)\). Then for \(\lambda\in\Lambda_{\eta,K}\),

(i)

if equation (1.1) has exactly one linearly independent solution satisfying (3.10) and this is the only linearly independent solution of equation (1.1) in \(L^{2}_{w}(\rho(0),+\infty)\), then equation (1.1) is called case I;

(ii)

if equation (1.1) has exactly one linearly independent solution satisfying (3.10), but all the solutions of equation (1.1) are in \(L^{2}_{w}(\rho(0),+\infty)\), then equation (1.1) is called case II;

(iii)

if all the solutions of equation (1.1) satisfy (3.10) and hence are in \(L^{2}_{w}(\rho(0),+\infty)\), then equation (1.1) is said to be in the case III.

Remark 3.1

It follows from Corollary 3.1 and (3.9) that case I and case II are the sub-cases of the limit-point case and case III is the limit-circle case.

The next theorem in this section shows that the classification of equation (1.1) into cases I, II, and III is independent of the choice of λ.

Theorem 3.3

(i)

If all the solutions of equation (1.1) are in\(L^{2}_{w}(\rho (0),+\infty)\)for some\(\lambda_{0}\in\mathbb{C}\), then the same is true for all\(\lambda\in\mathbb{C}\).

(ii)

If all the solutions of equation (1.1) satisfy (3.10) for some\(\lambda_{0}\in\Lambda_{\eta,K}\), then the same is true for all\(\lambda\in\mathbb{C}\).

Proof

(i) Suppose that equation (1.1) has two linearly independent solutions in \(L^{2}_{w}(\rho(0), +\infty)\) for \(\lambda =\lambda_{0}\in\mathbb{C}\). Then \(y_{1}(t,\lambda_{0})\) and \(y_{2}(t,\lambda_{0})\) are both in \(L^{2}_{w}(\rho(0),+\infty)\). For briefness, denote

It follows from (3.12) that \(v(\cdot)\in L^{2}_{w}(a,+\infty)\) and hence \(v(\cdot)\in L^{2}_{w}(\rho(0),+\infty)\). Therefore, all the solutions of (1.1) are in \(L^{2}_{w}(\rho(0),+\infty)\) for all \(\lambda\in\mathbb{C}\).

(ii) Suppose that equation (1.1) has two linearly independent solutions satisfying (3.10) for \(\lambda=\lambda_{0}\in\Lambda_{\eta,K}\). Then \(u_{1}(t)=y_{1}(t,\lambda_{0})\) and \(u_{2}(t)=y_{2}(t,\lambda_{0})\) also satisfy (3.10). For any \(\lambda\in\mathbb{C}\), let \(v(t)\) be an arbitrary non-trivial solution of (1.1), and \(u(t)\) be the solution of (1.1) with \(\lambda=\lambda_{0}\), which has the initial values \(u(a)=v(a)\), \(u^{\Delta}(a)=v^{\Delta}(a)\), \(a\in(0,+\infty )\cap \mathbb{T}\). It follows from the variation of constants, \(u(t)\) and \(v(t)\) also satisfy (3.11). Differentiating both sides of (3.11), we get

and hence \(v(t)\) satisfy (3.10). Therefore, all solutions of equation (1.1) satisfy (3.10) for all \(\lambda\in\mathbb{C}\). This completes the proof. □

Remark 3.2

(i)

Equation (1.1) is in the case I if it has a solution y not to be in \(L^{2}_{w}(\rho(0),+\infty)\). In fact, it can be concluded from \(y(\cdot)\notin L^{2}_{w}(\rho(0),+\infty)\) and (2.5) that y does not satisfy (3.10).

(ii)

By (i) of this remark and (i) of Theorem 3.3, if equation (1.1) is in the case I with respect to some \(\Lambda_{\eta_{0},K_{0}}\), then the same is true for all \(\Lambda_{\eta,K}\), that is, case I is independent of \(\Lambda_{\eta,K}\).

Remark 3.3

(i)

In the continuous case: \(\mu(t)\equiv0\). Theorem 3.1-3.3, Corollary 3.1, and Definition 3.1 are the same as those obtained by Brown et al. for formally non-self-adjoint second-order differential equations [19], Section 2.

(ii)

In the discrete case: All the points of \([\rho(0),+\infty )\cap\mathbb{T}\) are isolated. Let \([\rho(0),+\infty)\cap\mathbb {T}=\{t_{-1},t_{0},t_{1},\ldots\}\), where \(t_{-1}< t_{0}< t_{1}<\cdots\). In this case, equation (1.1) can be written as

where \(\tilde{p}(t_{n})=p(t_{n})/\mu(t_{n})\), \(\tilde{q}(t_{n})=q(t_{n})\mu (t_{n})\), and \(\tilde{w}(t_{n})=w(t_{n})\mu(t_{n})\). By setting \(x(n)=y(t_{n})\), \(\hat{p}(n)=\tilde{p}(t_{n})\), \(\hat{q}(n)=\tilde {q}(t_{n})\), and \(\hat{w}(n)=\tilde{w}(t_{n})\), the above problems can be rewritten as

It is evident that the above equation is of a form similar to (1.3). Hence, Theorems 3.1-3.3, Corollary 3.1, and Definition 3.1 in this special case are the same as Theorems 3.1-3.5 in [21].

It has been known that case I is independent of \(\Lambda_{\eta,K}\) by (ii) of Remark 3.2. Time scales contain two special cases: continuous case and discrete case. It follows from the former examples [44], Examples 3.1, 3.2 and [24], Examples 3.1, 3.2, that cases II and III are dependent on \(\Lambda_{\eta,K}\), that is, there exist \(\Lambda_{\eta _{1},K_{1}}\) and \(\Lambda_{\eta_{2},K_{2}}\) with \(\Lambda_{\eta_{1},K_{1}}\cap \Lambda_{\eta_{2},K_{2}}\neq\emptyset\) such that equation (1.1) is in the case II with respect to \(\Lambda_{\eta_{1},K_{1}}\) and case III with respect to \(\Lambda_{\eta_{2},K_{2}}\). Now, we give the exact dependent of case II and case III on the corresponding half-planes by the following results.

Theorem 3.4

Assume that\(\eta_{1}, \eta_{2}\in\mathscr{B}\)and\(\eta_{1}\neq\eta _{2}\) (modπ). If equation (1.1) is in the case III with respect to\(\Lambda_{\eta_{1},K_{1}}\)and\(\Lambda_{\eta_{2},K_{2}}\), respectively, then equation (1.1) is in the case III with respect to all\(\Lambda_{\eta,K}\).

Proof

Using \(\sin(\eta_{2}-\eta_{1})=\sin\eta_{2}\cos\eta_{1}-\cos\eta_{2}\sin \eta_{1}\) and \(\cos(\eta_{j}+\phi(t))=\cos\eta_{j}\cos\phi(t)-\sin \eta_{j} \sin\phi(t)\), \(j=1,2\), and noting that \(\sin(\eta_{2}-\eta_{1})\neq0\) by \(\eta_{2}\neq \eta_{1}\) (modπ), we have

Now, let \((\eta,K)\in S(\alpha)\) and \(\lambda\in\Lambda_{\eta ,K}\). Let \(u(t)\) be a solution of (1.1). Suppose that equation (1.1) is in the case III with respect to \(\Lambda_{\eta_{1},K_{1}}\) and \(\Lambda _{\eta_{2},K_{2}}\), respectively. Then, it follows from (ii) of Theorem 3.3 that \(u(t)\) satisfies (3.10) with η, K replaced by \(\eta _{1}\), \(K_{1}\) and \(\eta_{2}\), \(K_{2}\), respectively. Then, we can get from (2.5), (3.10), and \(u(\cdot)\in L_{w}^{2}(\rho(0),+\infty)\),

Note that \(\lambda\in\Lambda_{\eta,K}\). We then see from (3.20) and (2.3) that (3.10) holds for each solution \(u(t)\) of (1.1). Hence, equation (1.1) is in the case III with respect to \(\Lambda_{\eta,K}\). This completes the proof. □

Corollary 3.2

If equation (1.1) is in the case II with respect to an\(\Lambda_{\eta,K}\), then there exists at most one\(\eta_{0}\in\mathscr{B}\) (modπ) such that equation (1.1) is in the case III with respect to\(\Lambda_{\eta_{0},K_{0}}\).

4 Several criteria of the limit cases

Denote \(y^{\Delta^{\sigma}}(t):=y^{\Delta}(\sigma(t))\), \(y^{\sigma ^{\Delta}}(t):=(y^{\sigma}(t))^{\Delta}\), and \(y^{\sigma^{2}}(t):=y^{\sigma}(\sigma (t))\). In this section, we assume that \(y^{\Delta^{\sigma}}(t)=y^{\sigma ^{\Delta}}(t)\) for all \(t\in[\rho(0),+\infty)\cap\mathbb{T}\) in order to establish several criteria for equation (1.1) to be in the case I on the special time scales. It follows from \(y^{\sigma^{\Delta}}(t)=(1+\mu ^{\Delta}(t))y^{\Delta^{\sigma}}(t)\) and the assumption \(y^{\Delta ^{\sigma}}(t)=y^{\sigma^{\Delta}}(t)\) that \(\mu^{\Delta}(t)\equiv0\), that is, the graininess \(\mu(t)\) is constant, which implies \(\mathbb{T}=\mathbb {R}\) or \(\mathbb{T}=h\mathbb{Z}\). Since \(\mu(t)\) is a constant, the integrals \(\int^{+\infty}f(t)\Delta t\) and \(\int^{+\infty}f^{\sigma}(t)\Delta t\) are convergent or divergent simultaneously.

Theorem 4.1

Assume that\(\mathscr{B}\)contains at least two different elements\(\eta_{1}\)and\(\eta_{2}\) (modπ). If

Proof

Since \(\eta_{1}, \eta_{2}\in\mathscr{B}\) are different (modπ), the imaginary axes of the two half-planes \(\Lambda_{\eta _{1},K_{1}}\) and \(\Lambda_{\eta_{2},K_{2}}\) intersect. Therefore, we have \(\Lambda_{\eta_{1},K_{1}}\cap\Lambda_{\eta_{2},K_{2}}\neq\emptyset\). Choose \(\lambda\in\Lambda_{\eta_{1},K_{1}}\cap\Lambda_{\eta_{2},K_{2}}\). Let \(u(t)\) be the solution of equation (1.1) satisfying the initial value conditions

Choose \(t_{1}\in(\rho(0),+\infty)\cap\mathbb{T}\) and let \(h:=\min \{\delta_{1}\int_{\rho(0)}^{t_{1}}w(s)|u^{\sigma}(s)|^{2}\Delta s,\delta _{2}\int_{\rho(0)}^{t_{1}}w(s)|u^{\sigma}(s)|^{2}\Delta s \}\). Then we get from (4.2) and (4.3)

Now, we show that \(u(\cdot)\notin L^{2}_{w}(\rho(0),+\infty)\). Suppose on the contrary that \(u(\cdot)\in L^{2}_{w}(\rho(0), +\infty)\). Then it follows from (1.1), (2.1), (2.3), (2.5), (4.4), and the Schwarz inequality that

which yields \(\lim_{t\rightarrow+\infty}V(t)=+\infty\) by \(\int _{\rho(0)}^{+\infty}\sqrt{\frac{ (w(t)w^{\sigma}(t) )^{\frac {1}{2}}}{|p^{\sigma}(t)|}}\Delta t=+\infty\) and \(u(\cdot)\in L^{2}_{w}(\rho (0),+\infty)\). On the other hand, we get from (4.6)

which implies that \(\lim_{t\rightarrow+\infty}\frac {1}{V(t)}=-\infty\). Then we have a contradiction with \(\lim_{t\rightarrow+\infty}V(t)=+\infty\). So, \(u(\cdot)\notin L_{w}^{2}(\rho (0),+\infty)\) and equation (1.1) is in the case I by (i) of Remark 3.2. This completes the proof. □

Clearly, \(\mathscr{B}\) contains at least two different elements \(\eta _{1}\) and \(\eta_{2}\) (modπ). Further, it is clear that \(\int _{\rho(0)}^{+\infty}\sqrt{\frac{ (w(t)w^{\sigma}(t) )^{\frac {1}{2}}}{|p^{\sigma}(t)|}}\Delta t=+\infty\). So, (4.8) is in the case I by Theorem 4.1.

Proof

Clearly, \(\mathscr{B}\) contains at least two different elements \(\eta _{1}\) and \(\eta_{2}\) (modπ) since all the points \(x\geq\inf\{\frac {q(t)}{w(t)}\}\) form a half line in the complex plane \(xoy\). In addition, \(\int_{\rho(0)}^{+\infty}\sqrt{\frac{(w(t)w^{\sigma}(t))^{\frac{1}{2}}}{|p^{\sigma}(t)|}}\Delta t=\int_{\rho(0)}^{+\infty }\sqrt{(w(t)w^{\sigma}(t))^{\frac{1}{2}}}\Delta t=+\infty\). So, equation (1.1) is in the case I by Theorem 4.1. This completes the proof. □

In the following, denote \(p_{1}(t):=\Re[p(t)]\), \(q_{1}(t):=\Re[q(t)]\), \(p_{2}(t):=\Im[p(t)]\), and \(q_{2}(t):=\Im[q(t)]\). It is noted that Theorem 4.1 cannot be used for equation (1.1) for which there is only one element in \(\mathscr{B}\). So, we establish the following criterion which can be used in this case.

Theorem 4.2

Let\(p_{1}(t)>0\)for all\(t\in [\rho(0),+\infty)\cap\mathbb{T}\). If there exists a positive Δ-differentiable function\(M(t)\)on\([t_{0},+\infty)\cap\mathbb{T}\)for some\(t_{0}\in[\rho(0),+\infty)\cap\mathbb{T}\)and four positive constants\(k_{0}\), \(k_{1}\), \(k_{2}\), \(k_{3}\), such that for all\(t\in [t_{0},+\infty)\cap\mathbb{T}\):

Proof

Let \((\eta,K)\in S(\alpha)\) and \(\Lambda_{\eta,K}\) be the corresponding half-plane. Choose \(\lambda\in\Lambda_{\eta ,K}\). Let \(u(t)\) be the solution of (1.1) satisfying the initial value conditions (4.1). Then (4.2) and (4.3) hold with \(\eta_{j}\) and \(\delta _{j}\) replaced by η and δ, where δ is the distance from λ to \(\partial\Lambda_{\eta,K}\). Then it follows that there exist \(\tilde{t}_{1}\in(\rho(0),+\infty)\cap\mathbb{T}\) and a positive constant h̃ such that

Next, we show that \(u(\cdot)\notin L_{w}^{2}(\rho(0),+\infty)\). Assume the contrary. Suppose that \(u(\cdot)\in L_{w}^{2}(\rho(0),+\infty)\). Then, using the assumptions (ii), (iv), and the Schwarz inequality, we have

which, together with (4.9) and \(u(\cdot)\in L^{2}_{w}(\rho(0),+\infty)\), implies that \(\lim_{t\rightarrow+\infty}W(t)=+\infty\). It can be seen from \(M^{\sigma}(t)>k_{0}\), \(\lim_{t\rightarrow+\infty }W(t)=+\infty\), and (4.13) that there exists \(\tilde{t}_{2}\in (t_{2},+\infty)\cap\mathbb{T}\) such that

which implies that \(\lim_{t\rightarrow+\infty}\frac {1}{W(t)}=-\infty\) by (4.9). Then, we get a contradiction with \(\lim_{t\rightarrow+\infty}W(t)=+\infty\). So, \(u(\cdot)\notin L_{w}^{2}(\rho (0),+\infty)\) and equation (1.1) is in the case I by (i) of Remark 3.2. This completes the proof. □

Remark 4.1

Theorem 4.2 extended the related result Theorem 4.1 of [23] for second-order differential equation with complex coefficients to the time scales. In addition, let \(p^{\Delta}\), q be real functions on \([\rho(0),+\infty)\cap\mathbb{T}\), then Theorem 4.2 contains the criterion of the limit-point case for the formally self-adjoint systems, which is similar to Theorem 4.1 of [38].

It is noted that more limitations are imposed on \(\Re[p(t)]\) and \(\Re [q(t)]\) in Theorem 4.2. Integrating both sides of (4.11) and taking the imaginary part, we can get the following criterion.

By choosing \(M(t)=t\), it can be verified that (4.15) is in the case I by Theorem 4.2.

Declarations

Acknowledgements

Many thanks are due to Roman Šimon Hilscher (the editor) and the anonymous reviewers for helpful comments and suggestions. This research was supported by the NNSF of China (Grant 11571202) and the NSF of University of Jinan (XKY1511).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SS supervised the study and helped the revision. CZ carried out the main results of this article and drafted the manuscript. All the authors have read and approved the final manuscript.